b) Given F(r, y, z) = (x³ + cosh z) i+ (2y - 3ry)j – (x² + 4y²z) k. Use %3D Gauss's theorem to calculate F.n dS where n is the outward unit normal of o, the surface bounded by the planes, r 0, z = 0 ane r+2 = 6, and the parabolic cylinder r = 4 – y?.

Q: 4. Find the surface area of the paraboloid S defined by the equation: (r, y, z) = (u cos v, usin v,…

A: We will find surface area by using formula.

Q: Evaluate the surface integral. || x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

A: Given: ∫∫sx2yzdS where S is the part of the plane z=1+2x+3y that lies above the rectangle…

Q: 2. Evaluate the triple integral ///_y dV; G is the solid enclosed by the plane z = y, the xy-plane,…

A: To evaluate ∭GydV,  where G is the solid enclosed by the plane z=y, the x-y plane, and the parabolic…

Q: Consider the surface S= {(x, y,=)| xy + z =5; x20; y20; z20} in R. Use the Method of Lagrange to…

A: Given  surface S=x,y,x   ; xy+z=5   ;   x≥0 , y≥0  , z≥0 consider the point  x,y,z on the surface S…

Q: 3) Evaluate the surface integral. I| (x²z+ y²z) ds where S is the part of the plane z = 4 + x +y…

A:

Q: b) Given F(r, y, z) = (r³ + cosh z) i+ (2y3 – 3r*y)j – (2² + 4y²z) k. Use Gauss's theorem to…

A:

Q: . Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = y², xy-plane and…

A:

Q: b) Given F(x, Y, z) = (x³ + cosh z) i+ (2y³ – 3x²y)j – (x² + 4y²z) k. Use Gauss's theorem to…

A:

Q: If E is the solid shape bounded by x 2 0,y 20 ,z 2 0 and the plane 2x + 3y +z = 6 evaluate (x + y)…

A: The given question is solved below

Q: Q#5. Evaluate the surface integral ff(x,y,z)dS When f(x, y, z)= 3xyz and is the portion of the plane…

A:

Q: 5. Evaluate the surface integral S_(x+ y) ds ; o is the part of the plane 2x + y +z = 3 that lies in…

A:

Q: Find the surface area of the part of the plane 2x+2y +z=4 that lies inside the cylinder x²+y² = 2.…

A: Given that, the equation of the plane is 2x+2y+z=4 and the equation of the cylinder x2+y2=2. Note…

Q: Compute 3xy dS, where S is the surface of the tetrahedron with sides z = 0, y = 0, x + z = 1, and x…

A:

Q: b) Given F(.r, y. :) = (r + cosh 2)i+ (2y-3r*y)j-(r + 4y*:) k. Use Gauss's theorem to calculate F.n…

A:

Q: a) Let G be the wedge in the first octant that is cut from the cylindrical solid y² + z² ≤ 4 by the…

A: To Evaluate: ∫∫∫GxyzdV. Where G: y2+z2≤4, y=2x , x=0. As per policy, we are solving only thr first…

Q: Find by using polar coordinates in R^2 the volume of the region bounded by the parabolic z=…

A:

Q: Calculate ff F.n dS where F x'yzi + 2xj + y?k in the solid region S bounded by the parabolic tube z…

A:

Q: Use Stokes' theorem to calculate JJ (A) nds by A = 3yi-xzi+yz²k where S is the surface of the…

A: The given vector field is: A=3yi-xzj +yz2k The surface S is the paraboloid 2z=x2+y2 bounded by z=2.…

Q: Use cylindrical coordinates to evaluate the triple integral ∫∫∫E √(x2+y2)dV, where E is the solid…

A:

Q: (9) Evaluate the surface integral / S, czdS, where o is the part of the plane r+2y+3z = 6 that lies…

A:

Q: Calculate F. dS where F = (e" sin y, e² cos y, yz²) S is the surface of the box bounded by the…

A:

Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 O x - y -…

A:

Q: (4) Use Stokes' Theorem to evaluate 11. Vx (yz, xz, xy) dS where S is the surface S that is the part…

A:

Q: Consider the surface S={(x, y, z)| xy+ z= 5; x20; y20; z20} in R3. Use the Method of Lagrange to…

A: The distance between points can be calculated using the distance formula. The distance formula is…

Q: Use cylindrical coordinates to evaluate SI vz dv where E is the solid E bounded by the cylinder,…

A:

Q: Evaluate fff (3x + 2y)dV, where S is the solid bounded above by the plane z=4, below by plane z=0…

A: ∫∫∫(3x + 2y) dV=∫∫∫(3x + 2y) dxdydz = ∫z = 04∫y = -13∫x =-11(3x + 2y) dxdydz = ∫z = 04∫y = -134y…

Q: 4. Find the volume of the solid bounded by the plane z = 0 and the paraboloid z=1-x² - y². (if using…

A: Solution :-

Q: Evaluate SſF-dš where F = yi +2x j +(z-8)k and S is the surface of the solid bounded by 4x + 2y +z =…

A: By using the divergence theorem we solve the above question

Q: If V = (x +y) z, evaluate ∮ ? ??, where S is the surface of the cylindrical wedge defined by 0 < ϕ <…

A: Consider the given information. V=x+yz and the given limits are 0&lt;ϕ&lt;π2, 0&lt;z&lt;2. Convert…

Q: #3. Evaluate the surface integral xz ds, where S is the part of the plane 2x +2y +z = 4 that lies in…

A:

Q: Evaluate the surface integral. (z + x2y) ds S is the part of the cylinder y2 + z2 = 16 that lies…

A: Given ∫∫S(z+x2y)dS where S is the part of the cylinder y2+z2=16 that lies between the planes x=0 and…

Q: Calculate F- dS where F = (e" sin y, e* cos y, yz²) S is the surface of the box bounded by the…

A:

Q: b) Given F(x, y, z) = (x³ + cosh z) i+(2y³ – 3x²y) j – (x² + 4y²z) k. Use Gauss's theorem to…

A:

Q: 1. Consider the surface which is the part of the sphere r? + y? + 2? = 4 that lies above the cone z…

A:

Q: 10/ Compute e Fids, where S is the surface z =N16-x²-y² oriented up ward and F = <y,-x, zZ %3D 工=?

A:

Q: What is the equation of the plane to the surface z = e3Y sin(3x) +2 at the point P,0, 3) O x- y + ez…

A:

Q: b) Given F(r. y. :) = (r* + cosh : )i+(2y – 3.r"y) j - (r + 4y:) k. Use Gauss's theorem to calculate…

A:

Q: 15. Evaluate SLL:(z+1)dV where E is the solid enclosed by the paraboloid y = x2 + z² and the plane y…

A:

Q: Q4// A surface S is defined by x2 +z = 1 and the bounded by the planes y = 0,y = 2,x 0,z = 0 in the…

A:

Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 8x + 16y + 24z on the elipse…

A: To determine the maxima and minima of the function f(x, y, z) with respect to the function is…

Q: Evaluate the surface integral. JI (12 + y2 + 2) ds S is the part of the cylinder x2 + y2 = 9 that…

A:

Q: E) Sketch the solid bounded by the cylinder ** +y = 4 and the planes y+z = 4 and z = 0

A: Here, we need to sketch the solid bounded by the cylinder x2+y2=4 and the planes y+z=4 and z=0.

Q: #6. Use Stokes' Theorem to evaluate S. F dr, where F(x, y,2) = i+(x + yz)j+(xy-V) k and C is the…

A: Given that, Fx,y,z=i+x+yzj+xy-zk and C is the boundary of the part of the plane 3x+2y+z=1 in the…

Q: Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = the planes x = 0, x =…

A: Here I am using simple tripple integration to solve this question.

Q: Integrate G(x, y, z) = x2y2 + 4 over the surface cut from the parabolic cylinder y2 + 4z = 16 by the…

A:

Q: Use Stokes' Theorem to evaluate ∫CF⋅dr where F(x,y,z)=xi+yj+3(x2+y2)k and C is the boundary of the…

A:

Q: 1. Use the cylindrical coordinate to evaluate fll. x² + y²dV where G is the solid bounded by…

A:

Q: Calculate / F . dS where F = (e* sin y, e cos Y, yz²) S is the surface of the box bounded by the…

A:

Q: 10. Use cylindrical coordinates to evaluate SS S(x+y+z) dV where E is the solid enclosed by the…

A: The region